Topics in Coding Theory
Wednesdays at 14:30-16:30, Room 208 (Math 2)
We will cover advanced topics in coding theory focusing on properties and constructions beyond the basics of rate/distance/decoding-algorithms. We will see several notions of locality in codes that guarantee both resilience and usability. We will explore a variety of techniques and constructions of codes ranging from algebraic to combinatorial. Finally, we will see some applications of error-correcting codes both within TCS, and beyond.
Administrative stuff: Please sign up here for a lecture. Also, please sign up here to join the seminar's mailing list.
A basic course in coding theory, see here, or for example several chapters in this book.
Part I: Locally Decodable Codes (LDCs)
Locally decodable codes "LDCs" are codes that allow very quick random-access retrieval of information. Unlike classical codes, to retrieve a piece of information you do not need to read the entire codeword. We will use Yekhanin's survey paper as the main text.
Definitions, Reed-Muller, local decoding & params.
Matching Vectors codes
* PIR - Private Information Retrieval (Dvir-Gopi)
* Efremenko: from irreducible representations to LDCs
Part II: Constructions with expanders
Expander graphs are pseudorandom graphs that are sparse but behave as if they were dense. There are many ways in which expanders are applied towards constructing error-correcting codes. These have various desirable properties akin to LDCs: local correctability, local testability, symmetry, good rate/distance tradeoffs, and more.
- LCCs with n^\epsilon queries [HOW]
LCC, LTC via expanders [KORS, GKORS]
Ta-Shma’s Zigzag code [paper]
Spielman’s linear-time decoding of a code [paper]
Highly symmetric expander codes [paper]
Part III: Various topics
In the last part of the seminar we will explore various directions including applications of error-correcting codes in complexity, and in real-life applications (LRCs). We will take a look at quantum codes
Applications of coding theory:
- Hardness amplification and local list decoding
- PCPs, LTCs, and relaxed LDCs
Locally Repairable Codes (LRCs) - (lecture)
Yekhanin's survey paper
Venkat Guruswami's course notes.
Anup Rao's course notes
- Ta-Shma's course on P vs. BPP.